Bubble-type solutions of nonlinear singular problems

The paper describes the set of all solutions of the singular initial problems ( p ( t ) u ′ ) ′ = p ( t ) f ( u ) , u ( 0 ) = B , u ′ ( 0 ) = 0 , on the half–line [ 0 , ∞ ) . Here B < 0 is a parameter, p ( 0 ) = 0 and p ′ > 0 on ( 0 , ∞ ) , f ( L ) = 0 for some L > 0 and x f ( x ) < 0 if x < L , x ≠ 0 . By means of this result, the existence of a strictly increasing solution of this problem satisfying u ( ∞ ) = L is proved under some additional assumptions. In particular cases this homoclinic solution determines an increasing mass density in centrally symmetric gas bubbles which are surrounded by an external liquid with density L .